htaccess physcially redirect in case of 404 - .htaccess

Question

To start off I know how to do a simple redirect to a 404 when a page doesn't exist with.
ErrorDocument 404 /index.php
Is what I want to do is a physical redirect to the index page. So if a user types in mydomain.com/abc and that page doesn't exist it does an actual redirect TO the index page. I don't want the url to remain the same with the bad address. Yes I know its not a big deal but it bugs me. I could accomplish this by having a custom 404 page redirect to the homepage but I don't want this.
I've tried this with the [R] redirect flag and obviously it doesn't work.
Options -MultiViews
RewriteEngine On
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^(.*)$ $1.php [L]
RewriteRule ^(.*)$ /index.php [L,R]
Update
This is the error I get it rewrites the url to mydomain.com/index.php
I tried to remove the index.php with just / and it didn't redirect but gave the same error below.
The page isn't redirecting properly
Firefox has detected that the server is redirecting the request for this address in a way that will never complete.
This problem can sometimes be caused by disabling or refusing to accept
cookies.

Answer 1

Do this:
RewriteEngine on
RewriteBase /
RewriteCond %{REQUEST_URI} !(?:\.\w+)$ [NC]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ([^/]+)/?$ $1.php
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)$ / [L,R]
Explanation:
checks whether the file has extension:
RewriteCond %{REQUEST_URI} !(?:\.\w+)$ [NC]
If not, checks whether file is present:
RewriteCond %{REQUEST_FILENAME} !-f
If not a file, checks whether it is a folder which is present:
RewriteCond %{REQUEST_FILENAME} !-d
If not append .php. If / is present at the end, remove it and append .php
RewriteRule ([^/]+)/?$ $1.php
Checks whether the .php appended or whatever extension file is actually a file which is present:
RewriteCond %{REQUEST_FILENAME} !-f
check whether it is a directory:
RewriteCond %{REQUEST_FILENAME} !-d
If not, then it is a invalid file request and redirects it to /
RewriteRule ^(.*)$ / [L,R]

Answer 2

This should be your mod_rewrite based 404 handling:
# if it is not a file, directory or link
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-l
RewriteRule ^ index.php [L,R]
However I am not sure what are you doing in your first rule? Are you trying to add .php to each request URI?